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Arithmetic of random variables: adding constants to random variables, multiplying random variables by constants, and adding two random variables together AP Statistics B pp. 373-74 1 Pp. 373-74 are just plain hard • I don’t like the way they are written • They give you the conclusion, but don’t give you a sense of WHY the rule is what it is • This lecture gives you the derivation of the rules • You do not have to memorize the derivations, but if you understand them, you will understand why the rules are what they are 2 Outline for lecture • 3 basic ideas: – Adding a constant to a random variable (X+c) – Multiplying a random variable by a constant (aX) – Adding two random variables together (X+Y) • Being able to add two random variables is extremely important for the rest of the course, so you need to know the rules • Once you can apply the rules for μX+Y and σX+Y, we will reintroduce the normal model and add normal random variables together (go z-tables!) 3 Remember! • It may be useful to take notes, but this PowerPoint with the narration will be posted on the Garfield web site. • So will a version that does not have narration if you want a smaller file. • Different learning: classes like this that make the lectures available on line require different skills than classes where your notes are all you have. 4 Beginning concepts • Let’s look at the algebra behind adding, subtracting, and multiplying/dividing random variables. • Here, we will only examine addition and multiplication – Subtraction is simply adding the negative of the addend – Division is simply multiplying by the reciprocal of the divisor 5 Adding a constant to a random variable • The first thing we’ll try is adding a constant c to a random variable. • We will first calculate the mean, and then look at the variance • Remember that given the variance, we can always take its square root and obtain the standard deviation. 6 • E(X+c)=E(X)+c, where c=some real number For the next slides, we’re going to be expanding the series being summed, and then regrouping the variables and simplifying. 7 Expanding the series • Let’s expand without the sigma (adding) operator to keep the algebra neater. 8 Rewriting the equation • We can rewrite this as a series of individual fractions, since • Thus, 9 Regrouping the equation • Now, collect like terms: • Note that c/n in parenthesis appears n times • Now, rewrite this as a sum: 10 Var(X+c) • Var(X+c)=VarX. • We start with the basic definition for variation (VarX): • If we have add a constant c on to random variable X, we have Xi+c replacing Xi • Remember, the new mean is μX+c. 11 Substitute and rewrite the equation • So we substitute Xi-c for Xi, and μX+c for μX, to get: • Let’s again deal only with the numerator and expand the square: 12 Quite a mess, right? • Look at this: • You wanna simplify THAT????? • So let’s simplify it by NOT expanding the square. • Instead, what is (Xi+c)-(μX+c) equal to BEFORE we square it? 13 Simple, simple, simple • Distribute the subtraction operator over μX+c, and we should get: Xi+c-μX-c=Xi-μX • If we substitute Xi-μX into the numerator, we get our original definition of variation, i.e., 14 What about the standard deviation? • The fact that the VARIANCE does not change means the STANDARD DEVIATION does not change, either. • How come? Remember that • Since VAR does not change, the standard deviation also does not change when a constant is added to the random variable 15 What have we proven so far? • We have looked at the effect of adding a constant to a random variable X, i.e., using X+c • We have 3 conclusions for X+c: μX+c=μX+c σX+c=σX Var(X+c)=Var(X) 16 Multiplying a random variable by a constant • Now let’s see what happens when we MULTIPLY the random variable X by some constant a • Let’s look at the mean first: μaX. • We will substitute aX for X in the definition of the mean: 17 Expand and analyze • Again, let’s expand the Xi terms without the sigma: 18 Variance when the random variable is multiplied • Seeing what happens with the variance upon multiplication is similar to adding a constant: 19 Again, what about the standard deviation? • This derivation also explains why, when we multiply a random variable by a, the standard deviation is a multiple a of the standard deviation of the random variable. • Recall the definition of the standard deviation: 20 Standard deviation: substitute and solve • Substitute “aX” for X, and we get 21 Recap of conclusion for aX (multiplying the random variable by a constant • Once again, three conclusions: 22 Final approach: adding two random variables together • Let’s substitute in X+Y into our formulae to find out how they change – (Remember that X-Y can be recast as an addition problem X+(-Y), so we do not need a separate derivation for X-Y) 23 Calculating the mean when adding two random variables • We again start with the standard definition of the mean, except that we substitute “X+Y” for X: • Once again, calculating the mean is easy peasy. 24 Calculating the variance • The variance, of course, will be harder and messier. In fact, the derivation is so bad that you’ll have to accept this one on faith: Var(X±Y)=Var(X)+Var(Y) 25 What about standard deviations? • First, let’s derive them from the Var formula • Since • Therefore: , 26 Recap of adding two random variables together 27 Here endeth the lesson. • You are not responsible for these derivations, but I hope it helps to explain why the forms on pp.373-74 are what they are. 28